a Xi :hep- h/0111020 2 7 Jan 2003
DIAS-STP-01-16
hep- h/0111020
Oc obe 2002
Fuzzy Complex G assmannian Spaces
and hei S a P oduc s
B ian P. Dolan 1,a and Oli e Jahn 2,b
aDepa men o Ma hema ical Physics
NUI Maynoo h, Maynoo h, I eland
bDublin Ins i u e o Ad anced S udies
10 Bu ling on Road, Dublin 4, I eland
Abs ac
We de i e an explici exp ession o an associa i e s a p oduc on non-
commu a i e e sions o complex G assmannian spaces, in pa icula o he
case o complex 2-planes. Ou exp ession is in e ms o a ini e sum o
de i a i es. This gene alises p e ious esul s o complex p ojec i e spaces
and gi es a disc e e app oxima ion o he G assmannians in e ms o a
non-commu a i e algeb a, ep esen ed by ma ix mul iplica ion in a ini e-
dimensional ma ix algeb a. The ma ices a e es ic ed o ha e a dimension
which is p ecisely de e mined by he ha monic expansion o unc ions on he
commu a i e G assmannian, unca ed a a ini e le el. In he limi o in ini e-
dimensional ma ices we eco e he commu a i e algeb a o unc ions on he
complex G assmannians.
1bdolan@ hphys.may.ie
2[email p o ec ed]
1 In oduc ion
The e has ecen ly been much in e es in non-commu a i e geome y, [1, 2], bo h as
a no el di ec ion in s ing heo y [3] and as a new ool in quan um ield heo y [4]. In
he la e app oach he heo y is o mula ed on a “ uzzy” space, a se ies o disc e e
app oxima ions o a con inuous space- ime mani old M. The space o ields on each
app oxima ion is ini e-dimensional, so he uzzy space se es as a egula o , simila
o a la ice. In con as o he la e , he unca ion enjoys all symme ies o he
app oxima ed con inuous mani old. This has se e al use ul implica ions including
he absence o a doubling p oblem o chi al e mions [5, 6]. Topologically non- i ial
ield con igu a ions can be included and he chi al anomaly eme ges na u ally in his
app oach [7, 8].
In mo e de ail, a uzzy space is gi en by a se ies o ini e-dimensional algeb as
AL ha app oxima e he commu a i e algeb a o unc ions on Min he ollowing
way: each ALis iden i ied wi h a subse o unc ions on Mand he p oduc in AL
induces a (non-commu a i e) p oduc o unc ions in his subse , he so-called s a
p oduc ; in he limi L→ ∞, he subse should exhaus he se o all unc ions and
he s a p oduc go o e o he (commu a i e) poin wise p oduc o unc ions. In
he p esen case, he algeb as ALa e ull ma ix algeb as. The uzzy spaces a e hus
ma ix geome ies, which go o e o he usual con inuous mani old as he size o he
ma ix is aken o in ini y.
Non-commu a i e s a p oduc s a e known o exis o e e y Poisson mani old, in
pa icula o symplec ic mani olds [9]. S a p oduc s ealised by ini e-dimensional
ma ix algeb as ha e been cons uc ed in [10] o all homogeneous K¨ahle mani olds
p o ided a ce ain quan isa ion condi ion on he me ic is sa is ied. These algeb as
can also be ob ained using gene alised cohe en s a es [11] o he me hod o o bi s
o i educible ep esen a ions o Lie g oups [12, 13]. The ela ion be ween hese
app oaches has been discussed in [14] and ha o de o ma ion quan isa ion in [15,
16]. Fo he ela ion o Fou ie ans o ma ion on g oup space see [17]. In e ms
o unc ions on he mani old, he abo e o mula ions p o ide an exp ession o he
s a p oduc as an in eg al o e he analy ical con inua ion o he unc ions. This
is no e y con enien o explici calcula ions in non-commu a i e ield heo y. An
explici , local o mula in e ms o a ini e numbe o de i a i es is so a only known
o complex p ojec i e spaces [18] including he case o he uzzy 2-sphe e [19] al eady
ea ed in [20].
In his pape , we de i e an analogous o mula o uzzy complex G assmanni-
ans, ha is ini e ma ix geome y app oxima ions o he usual G assmannians,
GN
k∼
=U(N)/[U(k)×U(N−k)], which a e homogeneous spaces isomo phic o he
space o complex k-planes in CN. The ma ix geome ies consis o ma ices ac -
ing on he i educible ep esen a ion o SU(N) which is gi en by he L- old Young
p oduc o he ep esen a ion on an isymme ic k- enso s. As Lis inc eased he
con inuous G assmannian is eco e ed. The special case k= 1 equi es he L- old
symme ic p oduc o he undamen al ep esen a ion o SU(N), as in [18]. The
s a p oduc o he case k= 2 is cons uc ed explici ly, using globally well-de ined
bu o e -comple e co-o dina es on he G assmannian. The esul is exp essed as
a ini e sum o e mul iple de i a i es o he unc ions, which a e decomposed in o
1
i educible ep esen a ions o he s abili y g oup S[U(k)×U(N−k)] ac ing on he
angen space. I is shown ha he s a p oduc educes o he usual commu a i e
p oduc as L→ ∞. The pa icula case G4
2may be o some in e es in ield heo y
as i is a symplec ic mani old which has S4as a Lag angian sub-mani old. The co -
esponding ma ix geome ies can be iewed as non-commu a i e e sions o T∗S4,
he co- angen bundle o S4[21]. A s a p oduc on he complex G assmannians
as an in ini e sum o e de i a i es is known [22, 23]. Howe e , his o mula canno
be es ic ed o ini e-dimensional sub-algeb as and he e o e canno se e as a s a
p oduc on a uzzy app oxima ion o he mani old.
The layou o he pape is as ollows: in sec ion 2 we desc ibe he complex
G assmannians, GN
k, in e ms o p ojec ion ope a o s ac ing in CN, we in oduce
a se o global co-o dina es which a e o e -comple e and sa is y a se o quad a ic
cons ain s which ensu es ha hey indeed desc ibe GN
k; in sec ion 3 we analyse he
algeb a o unc ions in e ms o ep esen a ions o SU(N); in sec ion 4 we desc ibe
he ini e ma ix geome ies ha de ine he uzzy G assmannians GN
k,F; sec ion 5
gi es he cons uc ion o he s a p oduc o he special case k= 2, GN
2,F, while
sec ion 6 p esen s a conjec u e on he possible o m o k > 2 and some obse a ions
on he cons uc ion; sec ion 7 gi es a summa y and conclusions and some echnical
de ails a e elega ed o he appendices.
2 Complex G assmannian spaces
The complex G assmannian space GN
k∼
=U(N)/[U(k)×U(N−k)] = SU(N)/S[U(k)×
U(N−k)] can be ep esen ed as he space o all He mi ean ank-kp ojec o s Pac -
ing on CN. This is easily seen since any such p ojec o can be diagonalised by an
elemen o U(N) and he e is s ill a esidual adjoin ac ion o U(k)×U(N−k)
which lea es i in a ian . I will be con enien o desc ibe GN
kusing a edundan
se o globally well-de ined co-o dina es, ξA,A= 1,...,N2−1. Fi s we in oduce
o hono mal He mi ean gene a o s Ao he Lie algeb a o SU(N) sa is ying
A B=1
NδAB +1
√2(dABC+i ABC) C,(1)
whe e ABCa e he s uc u e cons an s o SU(N) and dABC he componen s o he
usual symme ic aceless enso . The Aa e no malised so ha ( A B) = δAB.
This allows us o pa ame e ise Pin e ms o N2−1 eal pa ame e s ξA,
P=k
N+ξA A.(2)
The condi ion ha Pis as p ojec o ansla es in o
ξAξA=k(N−k)
Nand 1
√2dABCξAξB=N−2k
NξC,(3)
and ξAnow pa ame e ise GN
kwhen hese condi ions a e imposed. This cons uc ion
embeds he G assmannian in he space o aceless He mi ian ma ices, which we
iden i y wi h RN2−1pa ame e ised by he un es ic ed ξA.
2
As discussed in de ail in e e ence [18] o he case o CPN−1≡GN
1, he complex
s uc u e and me ic a e encoded in a He mi ean p ojec o
KAB≡ P A(1 −P) B=1
2PAB+iJAB(4)
whe e JAB=√2 ABC ξCis he complex s uc u e on GN
kand PAB=PBA=−(J2)AB
(indices a e aised and lowe ed wi h he la me ic δA
Bo RN2−1). The ma ix KAB
p ojec s de i a i es wi h espec o ξAon o he holomo phic angen space o he
G assmannian when ac ing on he igh and on o he an i-holomo phic angen
space when ac ing on he le ; so ∇A≡KAB∂/∂ξBis a holomo phic de i a i e and
¯
∇B≡KAB∂/∂ξAan an i-holomo phic one. To see ha KABis a p ojec o we use
he comple eness ela ion o he gene a o s,
( A)i
j( A)k
l=δilδkj−1
Nδijδkl,(5)
o show ha
KAB B=P A(1 −P) and BKBA= (1 −P) AP.(6)
(These ela ions will p o e e y use ul in he ensuing analysis.) Examining he eal
and imagina y pa s o KABsepa a ely e eals ha P=−J2and PJ =JP =J.
This means ha Pi sel is a p ojec o on o he angen space o GN
k, wi h ank
2k(N−k). An impo an obse a ion o he ollowing analysis is ha he di e en ial
ope a o s KAB∂/∂ξBcommu e wi h he cons ain s (3), as hey mus do, since K
p ojec s on o he angen space. This can also be p o en using (6). I is his ac
ha allows us o use he global co-o dina es, a he han local co-o dina es, in he
inal di e en ial exp ession o he s a p oduc .
Co a ian de i a i es can be cons uc ed by p ojec ing de i a i es wi h espec
o he la coo dina es ξA o he angen space. Mul iple co a ian holomo phic
de i a i es a e hus de ined as
∇A1···∇An (ξ)≡KA1
B1···KAn
Bn∂B1KB2
C2∂C2···KBn
Cn∂Cn (ξ)(7)
whe e ∂A=∂/∂ξA. In ou case he e is a simpli ica ion because
KABKCD(∂BKDE) = 0 (8)
which ollows om he de ini ion (4) o Kand he comple eness ela ion (5). I
implies
∇A1···∇An (ξ) = KA1
B1···KAn
Bn∂B1···∂Bn (ξ).(9)
3 Ha monic analysis
In o de o ob ain a ini e-dimensional unca ion o he space o unc ions on he
G assmannian, which is compa ible wi h he symme ies, we decompose he space
o unc ions in o i educible ep esen a ions o he isome y g oup G=SU(N).
We hink o GN
kas he space G/H o ( igh ) cose s in Gwi h espec o H=
S[U(k)×U(N−k)], he subse o ma ices in U(k)×U(N−k) wi h uni de e minan .
3
Func ions on G/H can hus be conside ed as unc ions on G ha a e in a ian unde
he le ac ion o H. They ans o m unde Gacco ding o he igh ac ion.
The ull space o unc ions on Gis spanned by he ma ix elemen s DJ
MM′o
all i educible uni a y ep esen a ions Jo G. Decomposing in o i educible ep e-
sen a ions o H, we may w i e he i s componen index as M= (n, j, m) whe e j
labels he i educible ep esen a ions o H,m he co esponding componen s and n
dis inguishes copies o equi alen ep esen a ions. The le ac ion o His hen gi en
by
DJ
(n,j,m)M′(h−1g) = X
m′′
DJ
(n,j,m) (n,j,m′′)(h−1)DJ
(n,j,m′′ )M′(g).(10)
The H-in a ian ma ix elemen s a e hose o which he i s index co esponds o
he i ial ep esen a ion o H, labelled o ins ance by j=m= 0. The space o
unc ions on G/H is hus spanned by he ma ix elemen s DJ
(n,0,0) M′.
Unde he igh ac ion o G,
DJ
(n,0,0) M′(gg′) = X
M′′
DJ
(n,0,0) M′′ (g)DJ
M′′ M′(g′),(11)
so, o ixed Jand n, he DJ
(n,0,0) M′span he ec o space o he ep esen a ion J
o G. The space o unc ions on GN
k hus con ains all i educible ep esen a ions o
SU(N) ha con ain he i ial ep esen a ion upon es ic ion o S[U(k)×U(N−k)].
The mul iplici ies a e gi en by he mul iplici y o he i ial ep esen a ion in he
es ic ion.
We will desc ibe ep esen a ions o SU(N) by Young diag ams. These will be
deno ed by hei symbols J= [j1, j2,...,jN−1] whe e jideno es he numbe o
columns o heigh io he diag am.3The undamen al ep esen a ion, o ins ance,
has J= [1,0,...] and he adjoin one J= [1,0,...,0,1]. No e ha he complex
conjuga e ep esen a ion is gi en by J∗= [jN−1,...,j1]. In appendix C, we show
ha a ep esen a ion Jo SU(N) con ains he i ial ep esen a ion o S[U(k)×
U(N−k)] i and only i i appea s in he di ec p oduc
ML≡[0k−1, L, 0N−k−1]⊗[0N−k−1, L, 0k−1] (12)
o L≥n/N whe e n=Pijiis he numbe o boxes in he diag am Jand 0k−1
s ands o k−1 ze o en ies. The mul iplici y in he es ic ion is he same as ha
in he di ec p oduc . The MLsa is y M1⊂M2⊂ ···, so hey p o ide a hie a chy
o unca ions o he space o unc ions on he G assmannian.
The ep esen a ion [0k−1, L, 0N−k−1] is he Young p oduc o Lan i-symme ic
k- enso s, o ins ance o k= 2, L= 5. As an example, o N= 6 and
k= 2, he i s unca ion is
M1= [0,1,0,0,0] ⊗[0,0,0,1,0] = [0,0,0,0,0] ⊕[1,0,0,0,1] ⊕[0,1,0,1,0] ,
⊗= 1 ⊕ ⊕ (13)
3No e ha his symbol is di e en om he highes weigh ec o some imes used o desc ibe
a diag am, he en ies o he la e being he numbe o boxes in each ow.
4

he second is
M2= [0,2,0,0,0] ⊗[0,0,0,2,0] = [0,0,0,0,0] ⊕[1,0,0,0,1] ⊕[0,1,0,1,0]
⊕[2,0,0,0,2] ⊕[1,1,0,1,1] ⊕[0,2,0,2,0] .
⊗= 1 ⊕ ⊕
⊕ ⊕ ⊕
(14)
Al hough no needed in he ollowing, we would like o s a e o illus a ional
pu poses ha he ull ha monic analysis on GN
k( he “union” o all ML) is gi en by
he ep esen a ions
J=([m1,...,mk,0,...,0, mk,...,m1] i 2k < N ,
[m1,...,mk−1,2mk, mk−1,...,m1] i 2k=N , (15)
wi h mi= 0,1,..., each ep esen a ion occu ing once. The case 2k > N can be
ob ained by eplacing kby N−k, since GN
k∼
=GN
N−k. The ep esen a ions (15)
co espond o Young diag ams which can be ob ained by pu ing a diag am wi h
a mos k ows nex o i s conjuga e (which has a leas N−k≥kboxes in any
column), as can be e i ied o he examples gi en abo e. This esul is also de i ed
in appendix C.
4 Ma ix geome y
In he p e ious sec ion, we ha e ob ained a se ies o unca ions o he space o unc-
ions on he G assmannian. We will now see ha hese ca y a na u al p oduc .
The ep esen a ion con en MLo a gi en unca ion can in ac be ealised as an
algeb a o ma ices in a ep esen a ion Jo SU(N): since such ma ices ans o m
unde SU(N) by conjuga ion, hey o m he ep esen a ion space o J⊗J∗; so ML,
as in oduced in eq. (12), is equi alen o he space o ma ices in he ep esen a-
ion [0k−1, L, 0N−k−1]. Since he ma ix p oduc espec s he ac ion o SU(N), he
algeb a MLhas he same symme ies as GN
k.
In o de o ob ain he co esponding p oduc o ( unca ed) unc ions, we shall
now cons uc an injec i e map om ML o he space o unc ions on he G ass-
mannian which also espec s he g oup ac ion (an equi a ian map). This map
au oma ically p o ides he no ions o di e en ia ion and in eg a ion needed o he
cons uc ion o ac ions: one jus has o map he co esponding no ions o unc ions
back o ma ices. Equi a iance gua an ees ha hey a e compa ible wi h he un-
ca ion. The map will also p o ide a non-commu a i e p oduc o unc ions in he
image o ML, he s a p oduc . I he s a p oduc ends o he poin -wise p oduc
in he limi L→ ∞, we ha e succeeded in cons uc ing a uzzy GN
k. Since we will
es ic ou sel es o k= 2 in he ollowing sec ions, we p esen he map only o his
case. The gene alisa ion o o he alues o kshould be ob ious.
5
Fo GN
2 he basic building block will be he an i-symme ic ep esen a ion ,
co esponding o L= 1. The i s non- i ial unca ion o unc ions he e o e
equi es using [N(N−1)/2]×[N(N−1)/2] ma ices. A unc ion on GN
2is associa ed
wi h such a ma ix ˆ
Fby es ic ing he enso p oduc o he undamen al p ojec o
(2) o he an i-symme ic ep esen a ion ,
ρ≡(P ⊗P)a,(16)
and cons uc ing
F1(ξ) = [ρ(ξ)ˆ
F].(17)
Since Phas ank 2, ρhas ank 1: le he plane on o which Pp ojec s be spanned by
he ec o s ~ and ~w;ρ hen p ojec s on o he 1-dimensional subspace o he ep esen-
a ion space o spanned by he an i-symme ic p oduc o ~ and ~w (an explici p oo
is gi en in appendix B). A gene al unca ion equi es aking he L- old (Young)
p oduc [0, L, 0,...] = ···
··· o , which has dimension nN
L=(N+L−1)!(N+L−2)!
(N−1)!L!(N−2)!(L+1)! ,
and using nN
L×nN
Lma ices. Equa ions (13) and (14), o ins ance, show he decom-
posi ion o 15 ×15 and 105 ×105 ma ices as ha monics o G6
2. In he ollowing we
shall d op ailing ze os in symbols o ep esen a ions, so he abo e ep esen a ions
will be deno ed by [0, L]. The Young p oduc can be ob ained as a componen o
he symme ic enso p oduc , so a p ojec o can be cons uc ed by es ic ing he
L- old enso p oduc o ρ o he ep esen a ion [0, L],
ρL= (
L imes
z}| {
ρ⊗···⊗ρ)[0,L],(18)
(o cou se ρ1=ρ) and a unc ion can be associa ed wi h any nN
L×nN
Lma ix ˆ
Fby
FL(ξ) = [ρL(ξ)ˆ
F].(19)
Appendix B con ains a p oo ha he map (19) is injec i e.
The ma ix geome ies in oduced he e coincide wi h hose ob ained om com-
plex line bundles [10] o gene alised cohe en s a es [11], see [14, 15]. FLis usually
called he co a ian symbol o he ope a o ˆ
Fin hese o mula ions, and injec i i y
is well known and ollows om an analy ici y a gumen . The ela ion wi h cohe en
s a es will be discussed in some mo e de ail in appendix B.
5 S a p oduc on GN
2
Mul iplica ion o unca ed unc ions on he G assmannian can now be de ined using
ma ix mul iplica ion. The s a p oduc o wo unc ions, FL= (ρLˆ
F) and GL=
(ρLˆ
G), is ob ained om he ma ix p oduc h ough he map
(FL⋆ GL)(ξ) = ρL(ξ)ˆ
Fˆ
G.(20)
By cons uc ion his is an associa i e p oduc and i keeps wi hin he class o unc-
ions unca ed a le el L. Ou aim is o ind an explici exp ession o his s a
6
p oduc , pu ely in e ms o FLand GLand hei de i a i es, hus elimina ing he
explici e e ence o ma ices.
By o hono mali y o he ma ix elemen s in he ep esen a ion [0, L],
Rdµ(g)D[0,L]
M1M2(g−1)D[0,L]
M3M4(g) = (1/nN
L)δM1M4δM2M3,ˆ
Fcan be expanded as
ˆ
F=Zdµ(g)˜
F(g)D[0,L](g) (21)
wi h ˜
F(g)≡nN
L D[0,L](g−1)ˆ
F.(22)
Inse ing his in o (19), we ob ain
FL(ξ) = Zdµ(g)ωL(ξ, g)˜
F(g) (23)
wi h
ωL(ξ, g)≡ ρL(ξ)D[0,L](g)(24)
and he s a p oduc can be exp essed as
(FL⋆ GL)(ξ) = Zdµ(g)Zdµ(g′)ωL(ξ, gg′)˜
F(g)˜
G(g′).(25)
We seek an exp ession o he s a p oduc in e ms o de i a i es ac ing on FL(ξ)
and GL(ξ). By eqs. (25) and (23), his can be achie ed by de i ing an exp ession
o ωL(ξ, gg′) in e ms o de i a i es o ωL(ξ, g) and ωL(ξ, g′) wi h espec o ξ. The
la e is g ea ly acili a ed by he obse a ion ha ωLcan be exp essed in e ms o
ω1,
ωL(ξ, g) = [ω1(ξ, g)]L,(26)
because, by eq. (18), ρL ac o ises in o ank-1 p ojec o s ρand D[0,L]ac s as a di ec
p oduc as well. The eason behind his ela ion is ha he ep esen a ion [0, L]
when p ojec ed o S[U(2) ×U(N−2)] by ρL ac o ises as [0,1]Lsince all o he
i educible componen s o he p oduc in ol e enso s ha a e an i-symme ic in 3
o mo e indices and he e o e anish in SU(2).
As a i s s ep, we ha e o ind an exp ession o ω1(ξ, gg′). This is a s aigh -
o wa d bu somewha leng hy exe cise. I is de e ed o appendix A and yields
ω1(ξ, gg′) = ω1(ξ, g)1 + ←−
∂AKAB−→
∂B+1
4←−
∂A←−
∂BKACKBD−→
∂C−→
∂Dω1(ξ, g′) (27)
whe e ∂A=∂/∂ξAand Kis he p ojec o on o he holomo phic angen space
in oduced in eq. (4). Subs i u ing (27) in (25) and in e changing di e en ia ion
wi h espec o ξwi h in eg a ion o e gand g′, we now ha e he s a p oduc a
le el one,
(F1⋆ G1)(ξ) = F1(ξ)1 + ←−
∂AKAB−→
∂B+1
4←−
∂A←−
∂BKACKBD−→
∂C−→
∂DG1(ξ).(28)
7
Fo highe Lwe ha e o conside ωL= (ω1)L. Equa ion (27) implies
ωL(ξ, gg′) = X
n+m≤L
L!
n!m! (L−n−m)! (ωω′)L−n−m(∂Aω)KAB(∂Bω′)n
×1
4(∂C∂Dω)KCEKDF (∂E∂Fω′)m
(29)
whe e we ha e used he abb e ia ions ω≡ω1(ξ, g) and ω′≡ω1(ξ, g′). The igh -
hand side o his equa ion has o be exp essed in e ms o mul iple de i a i es ac ing
on ωL(ξ, g) and ωL(ξ, g′). I con ains se e al di e en e ms wi h a gi en numbe
o de i a i es. This means ha we ha e o dis inguish componen s o mul iple
de i a i es o ωL.
To his end, we decompose mul iple holomo phic de i a i es ∇A1···∇AnωLas
de ined in eq. (9) wi h espec o i educible ep esen a ions o he s abili y g oup
Hwhich ac s on he angen space. I will be su icien o conside he subg oup
H0=SU(2) ×SU(N−2) o H. Rep esen a ions o H0will be deno ed by (J, J′)
whe e Jis a ep esen a ion o he i s ac o and J′one o he second. To ind he
ep esen a ion con en o a single holomo phic de i a i e ∇A=KAB∂B, no e ha
he undamen al ep esen a ion [1] o SU(N) decomposes as
[1]H0= ([1],[0]) ⊕([0],[1]) (30)
in o he undamen al ep esen a ions o SU(2) and SU(N−2) upon es ic ion o
H0. The wo componen s can be ob ained by p ojec ion wi h Pand 1 −P. Now
use eqs. (4) and (5) o w i e ∇Ain e ms o (an i-) undamen al indices,
( A)ij∇A (ξ) = (1 −P) BPi
j∂B (ξ).(31)
The ma ix B∂B ans o ms like he aceless componen o [1]×[1]∗unde SU(N).
Since he index iis p ojec ed by 1 − P o ([0],[1]) while jis p ojec ed by P
o ([1],[0])∗, we ind ha he holomo phic de i a i e ans o ms like ([0],[1]) ⊗
([1],[0])∗= ([1]∗,[1]). No e ha acelessness is gua an eed by he p ojec ions in
eq. (31). By he same easoning, an an i-holomo phic de i a i e ¯
∇A=KBA∂B
ans o ms like ([1],[1]∗). A mul iple holomo phic de i a i e ans o ms like he
symme ic enso p oduc o ncopies o he ep esen a ion ([1]∗,[1]). In o de o
ob ain an explici exp ession o he decomposi ion o his p oduc , i u ns ou
o be use ul o i s decompose he enso p oduc o ncopies o he undamen al
ep esen a ion o SU(N).
This can be done by conside ing he ac ion o he symme ic g oup Sn, whose
elemen s pe mu e he ac o s in he enso p oduc [28]. The la e can hen be
decomposed in o i educible ep esen a ions o SU(N)×Snwi h he help o cha ac e
p ojec ion ope a o s. They p o ide he ollowing decomposi ion o uni y,
1 = X
|J|=n
PJwhe e PJ≡dJ
n!X
π∈Sn
χJ(π)π . (32)
He e, he sum is o e all Young diag ams wi h nboxes, χJis he cha ac e o he
symme ic g oup in he ep esen a ion Jand dJ he dimension o ha ep esen a ion.
8
Since ρ1= (P ⊗P)ais a ank-1 p ojec o , he simple p oduc can be w i en as
ω(ξ, g)ω(ξ, g′) = (P ⊗P)a(g⊗g)a(P ⊗P)a(g′⊗g′)a.(64)
Using 1 = (P ⊗ P)a+ (P ⊗ (1 − P))a+ ((1 − P)⊗ P)a+ ((1 − P)⊗(1 − P))a,
eqs. (62), (63) and (64) can be combined o
ω(ξ, gg′) = ω(ξ, g)1 + ←−
∂AKAB−→
∂B+1
4←−
∂A←−
∂BKACKBD−→
∂C−→
∂Dω(ξ, g′),(65)
which is he desi ed exp ession.
B Cohe en s a es
We show ha he p ojec o ρLas gi en in eq. (18) has ank 1, and we p esen a
simple a gumen o why he map om ma ices o unc ions is injec i e.
To his end, we equi e a mo e explici ep esen a ion o ρL. The ec o space
o he i educible ep esen a ion o SU(N) wi h symbol J= [0, L] can be ealised
as a sub-space wi h ce ain symme y p ope ies o he space o 2L-index enso s.
We cons uc i as he image o a Young symme ise . We i s assign enso indices
o he boxes in he Young diag am o he ep esen a ion by pu ing he numbe s
1,2,...,2Lin ascending o de in o one column a e he o he , o ins ance 1 3 5 7
2468
o [0,4]. The Young symme ise is now de ined as
Y[0,L]=2L
L+ 1ALSL,
AL=
L
Y
i=1 1
2(1 −τi),SL=1
L!X
π1∈R1
π1
1
L!X
π2∈R2
π2
(66)
whe e τiin e changes he wo boxes o he i h column o he diag am and Rideno es
he se o pe mu a ions ha pe mu e he boxes o ow i. So SLsymme ises he
ows o he diag ams, while ALan i-symme ises he columns. Bo h a e symme ic
p ojec o s. The Young symme ise is a p ojec o , Y2
[0,L]=Y[0,L], bu no symme ic.
Ope a o s in he ec o space o [0, L] can be unambiguously desc ibed as ope -
a o s ˆ
Fon 2L- enso s ha sa is y
ˆ
FY[0,L]=Y[0,L]ˆ
F=ˆ
F . (67)
Now we can p o e ha ρLhas ank 1 by exp essing he ank-2 p ojec o Pin
e ms o an o hono mal basis |ϕi,|ψio he complex plane on o which i p ojec s,
P=|ϕihϕ|+|ψihψ|.
The le el-1 p ojec o ρwas de ined in (16) as he p ojec ion o he enso p oduc
o Pwi h i sel o he an i-symme ic ep esen a ion [0,1]. Since Y[0,1] educes o a
single an i-symme isa ion,
ρ≡(P ⊗P)a= (P ⊗P)Y[0,1] =|ϕψihϕψ|(68)
15

whe e
|ϕψi ≡ 1
√2|ϕi|ψi−|ψi|ϕi.(69)
Fo he p ojec o a le el L, we ob ain
ρL= (ρ⊗···⊗ρ)Y[0,L]=|ϕψiLhϕψ|LY[0,L](70)
The s a e |ϕψicomple ely cha ac e ises he plane ha co esponds o a poin in GN
2.
I is he e o e na u al ha i occu s as a undamen al objec in he cons uc ion.
The s a es |ϕψiLcoincide, up o a con en ional phase, wi h he gene alised cohe en
s a es discussed in [11]. Since ˆ
FY[0,L]=Y[0,L]ˆ
F,
FL(ξ) = hϕψ|Lˆ
F|ϕψiL.(71)
So FLis he co a ian symbol, as de ined in [11], o he ope a o ˆ
F.
This exp ession can be used o show ha he map is injec i e. We ha e o show
ha ˆ
Fcan be econs uc ed om FL. Since |ϕψiL= 2L/2AL(|ϕi|ψi)L,
FL(ξ) = 2Lhϕ|hψ|LALˆ
FAL|ϕi|ψiL.
Due o he an i-symme isa ion be ween |ϕiand |ψi his unc ion can be homo-
geneously ex ended o gene al (non-o hono mal) |ϕiand |ψi. We choose |ϕi=
PN
n=1 an|niand |ψi=PN
n=1 bn|niwi h canonical basis ec o s |ni. By di e en-
ia ing wi h espec o a,band hei complex conjuga es, all ma ix elemen s o
SLALˆ
FALSLcan be ob ained. Using he symme y (67), we ob ain SLˆ
Fand hus
also 2L
L+1ALSLˆ
F=ˆ
F.
C Res ic ions and di ec p oduc s
We shall de i e he ela ion be ween he es ic ion o ep esen a ions o G=SU(N)
o H=S[U(k)×U(N−k)] and he di ec p oduc o ce ain ep esen a ions
used in sec ion 3. In his appendix, we will allow o columns o heigh N,J=
[j1, j2,...,jN], in diag ams desc ibing ep esen a ions o SU(N). These do no lead
o new ep esen a ions, since ep esen a ions di e ing only by jNa e uni a ily equi -
alen , bu his gene alisa ion will make o mulas much simple . We embed Hin o
SU(N) as ei(N−k)ϕU′0
0 e−ikϕU′′(72)
whe e U′∈SU(k) and U′′ ∈SU(N−k) and eiϕ∈U(1). This shows ha H=
[SU(k)×SU(N−k)×U(1)]/Znwhe e nis he leas common mul iple o kand
N−k. Rep esen a ions o Hcan hus be conside ed as ep esen a ions o SU(k)×
SU(N−k)×U(1) ha ep esen Zn i ially. This ixes he cha ge qo he U(1)
ac o eiqϕ o he ep esen a ion modulo n. We will deno e hese ep esen a ions as
(J′, J′′)qwhe e J′and J′′ a e symbols o SU(k) and SU(N−k) ep esen a ions,
espec i ely, and qis he cha ge o he U(1) ep esen a ion.
16
The es ic ion o an SU(N) ep esen a ion o Hcan be w i en as
JH=M
J′,J′′
mJ
J′,J′′ (J′, J′′)(N−k)|J′|−k|J′′|.(73)
He e, |J|=Pijiis he numbe o boxes in he diag am Jand we assume ha he
diag ams ha e been chosen such ha he o al numbe o boxes in J′and J′′ is he
same as in J,
|J|=|J′|+|J′′|.(74)
No e ha he U(1) ep esen a ion is de e mined by he SU(k)×SU(N−k) ep esen-
a ion, so he mul iplici ies mJ
J′,J′′ a e he same as o he es ic ion om SU(N)
o he la e . I is known ([24, 25], also see [26, 27]) ha hese can be ob ained
om he decomposi ion o he di ec p oduc o he SU(N) ep esen a ions wi h
diag ams J′and J′′,
J′⊗J′′ =M
J
mJ
J′,J′′ Jin SU(N). (75)
He e, again, he es ic ion (74) on he numbe o boxes applies. No e ha in eq. (75)
J′and J′′ a e in e p e ed as SU(N) ep esen a ions while hey a e in e p e ed as
SU(k) espec i ely SU(N−k) ep esen a ions in eq. (73).
We a e in e es ed in he case whe e he i ial ep esen a ion o Happea s
on he igh -hand side o (73). This means ha J′and J′′ only ha e columns o
heigh kand N−k, espec i ely, J′= [0k−1, L′,0N−k−1] and J′′ = [0N−k−1, L′′,0k−1]
whe e 0k−1s ands o k−1 ze o en ies, e c. In addi ion, he U(1) cha ge has o
anish, (N−k)|J′|=k|J′′|. Since |J′|=L′kand |J′′|=L′′(N−k), his implies
L′=L′′ ≡L, so ha J′and J′′ a e complex conjuga e ep esen a ions o SU(N).
We conclude ha a ep esen a ion o SU(N) con ains he i ial ep esen a ion o
S[U(k)×U(N−k)] i and only i i appea s in he decomposi ion o he di ec
p oduc
ML≡[0k−1, L, 0N−k−1]⊗[0N−k−1, L, 0k−1] (76)
o some L. The mul iplici ies a e gi en by he mul iplici ies mJ
[0k−1,L,0N−k−1],[0N−k−1,L,0k−1]
in he p oduc . No e ha he mul iplici y does no depend on he numbe jNo
columns o heigh Nin he diag am Jchosen o a gi en ep esen a ion. This means
ha a ep esen a ion appea s in MLi and only i NL ≥ |J|whe e Jis he minimal
diag am (jN= 0) o he ep esen a ion. The e o e ML⊂ML′i L < L′.
Decomposi ion o he di ec p oduc
Fo illus a ional pu poses, we will explici ly pe o m he decomposi ion o he di ec
p oduc (76) in o i educible ep esen a ions. We can assume k≤N−ksince
GN
k∼
=GN
N−k. The decomposi ion is achie ed by Young diag am echniques. Recall
he ules o decomposing he di ec p oduc o wo i educible ep esen a ions o
SU(N) [28]:
1. Label each box in he second diag am by i s ow numbe .
17
nN−k+1 boxes →
nN−k+2 boxes →
nN−k+kboxes →
1·· ·· ·· ·· ·· ·· ·· ·· ·· 1
2·· ·· ·· ·· ·· ·· 2
·· ·· ·· ·· ·· ·· ··
·· ·· ·· ·· ·· ··
k·· k
1·· 1 2 ·· 2·· ·· k·· k
2·· 2·· ·· ·· ·· ··
·· ·· ·· ·· ·· ·· ··
·· ·· ·· ·· ·· ··
k·· k
←n11’s
←n22’s
←nkk’s
Figu e 1: Decomposi ion o he enso p oduc [0N−k−1, L, 0k−1]⊗[0k−1, L, 0N−k−1].
2. A ach all boxes wi h 1’s o he i s diag am, hen all boxes wi h 2’s and so
on, such ha
(a) a all s ages he in e media e diag am co esponds o an i educible ep-
esen a ion o SU(N), i.e. all columns s a in he i s ow and a e con-
nec ed and he heigh o he columns mono onically dec eases om le
o igh ,
(b) no column con ains any numbe mo e han once, and
(c) when coun ed om he igh , he n- h idoes no appea be o e he n- h
i−1.
Applying hese ules o ou case, we ha e o a ach boxes wi h Lcopies o each o
he numbe s 1,...,k o a ec angle o heigh N−kand wid h L. This is indica ed in
igu e 1. We ha e al eady an icipa ed some ac s abou he esul ing dis ibu ion
o boxes, which we will explain now. The numbe o 1’s in he i s ow has been
deno ed by n1. The emaining L−n11’s ha e o be in he N−k+ 1-s ow.
Deno ing he numbe o 2’s in he second ow by n2, he e can be a mos n1−n2
2’s in ow N−k+ 1, because o ule 2c. The e mus hus be a leas L−n12’s in
ow N−k+ 2. Howe e , owing o ule 2b, he numbe o 2’s in ha ow is a mos
L−n1. The e o e, he e ha e o be exac ly n1−n22’s in ow N−k+ 1 and L−n1
2’s in ow N−k+ 2. By he same easoning, we ind ha he numbe o i’s in ow
N−k+l o l > 0 is equal o he numbe o i−l+ 1’s in he ow N−k+ 1 which
is in u n equal o ni−l−ni−l+1 i i≥land 0 i i < l (we ha e pu n0≡L). Adding
up, we ind o he numbe o boxes in ow N−k+l,
nN−k+l=
k
X
i=l
(ni−l−ni−l+1) = L−nk−l+1 o l= 1,...,k. (77)
G aphically, his means ha he subdiag am o columns L+ 1, L + 2,...combines,
a e a o a ion by π, wi h he emainde o he diag am o a ec angle o wid h
Land heigh N. In e ms o symbols J= [j1,...,jN], whe e jiis he numbe o
columns o heigh i, his implies
J=([m1,...,mk,0,...,0, mk,...,m1] i 2k < N ,
[m1,...,mk−1,2mk, mk−1,...,m1] i 2k=N(78)
18
wi h mi=ni−ni+1 whe e nk+1 ≡0. All non-nega i e alues o misa is ying
Pk
i=1 mi=n1≤Loccu . Fu he mo e, each diag am can be ob ained in only one
way, since i is de e mined by he numbe s ni(i= 1,...,k). So all mul iplici ies
equal 1.
D Symme ic g oup
In his appendix, we shall p o ide a p oo o he ac o isa ion p ope y (38). On
he way, we will ecall some ac s abou ep esen a ions o he symme ic g oup and
he associa ed p ojec o s PJused in he ex . These p ojec o s can be conside ed
as elemen s o he g oup algeb a R[Sj]≡span Sn={A=Pπ∈SnAππ|Aπ∈R}, he
se o o mal linea combina ions o g oup elemen s. The ec o space R[Sj] ca ies
a ep esen a ion o he algeb a R[Sj] whose ac ion is gi en by le mul iplica ion.
This ep esen a ion es ic s o a ep esen a ion o he subg oup Sno R[Sj]. I
is usually called he egula ep esen a ion. Some imes i is con enien o iden i y
R[Sj] wi h he algeb a F(Sn) o unc ions on he g oup Snby se ing A(π)≡Aπ.
The ac ion o a g oup elemen πis hen gi en by (πA)(σ) = A(π−1σ). F(Sn) can in
ac be conside ed as he dual ec o space o R[Sj] since each unc ion on he g oup
can be linea ly and uniquely ex ended o a unc ion on R[Sj]. The iden i ica ion o
R[Sj] wi h i s dual space can be ob ained om he inne p oduc
hA, Bi ≡ X
π∈Sn
AπBπ=1
n! (ATB) (79)
by pu ing A(B) = hA, Bi. The ace in eq. (79) is o e he egula ep esen a ion
and we ha e se AT=PπAππ−1.
O pa icula impo ance a e he cen al elemen s o R[Sj], ha a e in a ian
unde conjuga ion wi h any g oup elemen π∈Sn,A=πAπ−1. They co espond
o class unc ions, i.e. unc ions ha depend only on he conjugacy class o hei
a gumen . An o hogonal basis in he subspace o class unc ions is gi en by he
cha ac e s χJassocia ed wi h he i educible ep esen a ions Jo Sn,
hχJ, χJ′i=n!δJJ′.(80)
So e e y class unc ion can be expanded as
A=X
J
AJχJwi h AJ=1
n!hχJ, Ai.(81)
To each A∈R[Sj], one can associa e a cen al elemen Aby a e aging wi h espec
o conjuga ion,
A≡1
n!X
π∈Sn
πAπ−1=1
n!X
JhχJ, AiχJ(82)
whe e we ha e used (81) and he in a iance o χJunde conjuga ion.
The egula ep esen a ion is in gene al educible. I con ains each i educible
ep esen a ion Jwi h a mul iplici y ha is gi en by he dimension dJo he ep e-
sen a ion. The componen con aining all copies o an i educible ep esen a ion J
19
can be ob ained as he image o he symme ic p ojec ion ope a o in oduced in
eq. (32), in he dual pic u e,
PJ=dJ
n!χJ.(83)
The decomposi ion o uni y
1 = X
J
PJ(84)
p o ides a decomposi ion o R[Sj] in o o hogonal subspaces [29].
Now we will show how he a e aged enso p oduc o wo p ojec o s PJ1and
PJ2on o i educible ep esen a ions o Sn1and Sn2can be exp essed in e ms o
i educible p ojec o s. PJ1⊗PJ2can be ex ended o R[Sj] whe e n=n1+n2. By
(82), we ha e
PJ1⊗PJ2=X
J
aJχJ(85)
wi h
aJ=1
n!hχJ, PJ1⊗PJ2i.(86)
The es ic ion o χJ o (π, σ)∈Sn1×Sn2decomposes in o i educible cha ac e s
as
χJ(π, σ) = X
J1,J2
cJ
J1J2χJ1(π)χJ2(σ) (87)
whe e cJ
J1J2∈Za e mul iplici ies o Clebsch-Go dan coe icien s. Wi h (80) we ge
aJ=1
n!X
J′
1,J′
2
cJ
J′
1J′
2hχJ′
1, PJ1ihχJ′
2, PJ2i=dJ1dJ2
n!cJ
J1J2(88)
and he e o e
PJ1⊗PJ2=X
J
dJ1dJ2
n!cJ
J1J2χJ=X
J
dJ1dJ2
dJ
cJ
J1J2PJ.(89)
By i e a ion, his esul can be gene alised o mul iple p oduc s,
1
dJ1···dJm
PJ1⊗···⊗PJm=X
J
cJ
J1...Jm
1
dJ
PJ.(90)
No e ha symme isa ion wi h espec o Snimplies symme isa ion wi h espec
o Sn′⊂Sn,
A⊗B⊗C=A⊗B⊗C . (91)
Now we can p o e eq. (38). The igh -hand side o his equa ion can be w i en as a
single ace like in eq. (36) bu wi h PJ eplaced by P[l]⊗P⊗m
[0,1]. Owing o he sym-
me ic enso s Sand Tall ac o s in he ace excep P[l]⊗P⊗m
[0,1] a e symme ic unde
conjuga ion, so P[l]⊗P⊗m
[0,1] can be eplaced by i s symme ised e sion P[l]⊗P⊗m
[0,1].
Now we can inse eq. (90). The only e m on he igh -hand side ha does no
anish when p ojec ed by P⊗n o a ep esen a ion o SU(2) is J= [l, m, 0,...] wi h
mul iplici y 1. The dimensions o he ep esen a ions [l] and [0,1] (o he symme ic
g oup) a e 1, while he dimension o [l, m] appea ing in he denomina o o (90) jus
cancels ha on he igh -hand side o (38), so we ob ain he le -hand side.
20

E P ojec ion o mul iple de i a i es
We compu e he p oduc o mul iple (an i-)holomo phic de i a i es o ωLand ω′
L
p ojec ed o he ep esen a ion ([l, m]∗,[l, m]) o he s abili y g oup,
X(L)
l,m ≡(∂A1···∂Al+2mωL)KA1...Al+2m,B1...Bl+2m
[l,m](∂B1···∂Bl+2mω′L).(92)
By eq. (9) and since K[l,m]con ains he p ojec o K, he de i a i es in his equa ion
a e eally co a ian de i a i es, holomo phic ones ac ing on ω′and an i-holomo phic
ones on ω. Equa ion (38) implies ha K[l,m]can be eplaced by d[l,m]K[l]⊗K⊗m
[0,1],
X(L)
l,m =d[l,m](∂l+2mωL)K[l]⊗K⊗m
[0,1](∂l+2mω′L) (93)
whe e we ha e in oduced an index- ee no a ion. Using he second equali y o
eq. (41), we ind
KAB,CD
[0,1] ∂C∂DωL=L(L+ 1)
2ωL−1KAB,CD
[0,1] ∂C∂Dω(94)
which i e a es o
(K[0,1]∂∂)mωL=L!(L+ 1)!
(L−m)!(L+ 1 −m)! ωL−m1
2K[0,1]∂∂ωm(95)
because he iple de i a i e o ω anishes. The i s equali y o eq. (41) implies ha
K[l]∂lωncon ains only single de i a i es o ω, whence
K[l]⊗K⊗m
[0,1]∂l+2mωL=K[l]⊗K⊗m
[0,1]∂l(K[0,1]∂∂)mωL
=L!(L+ 1)!
(L−l−m)!(L+ 1 −m)! ωL−l−mK[l](∂ω)l1
2K[0,1](∂∂ω)m(96)
whe e, in he i s s ep, we ha e used eq. (8). Since a simila equali y holds o
an i-holomo phic de i a i es, and K[l]and K[0,1] a e p ojec o s, we ind
X(L)
l,m =d[l,m]L!(L+ 1)!
(L−l−m)!(L+ 1 −m)!2
(ωω′)L−l−m
×(∂ω)lK[l](∂ω′)l1
4(∂∂ω)K[0,1](∂∂ω′)m.
(97)
Re e ences
[1] A. Connes, Noncommu a i e Geome y, Academic P ess (1994).
[2] J. Mado e, An In oduc ion o Noncommu a i e Di e en ial Geome y and
i s Physical Applica ions, Camb idge (1995); “Noncommu a i e geome y o
pedes ians,” g -qc/9906059.
[3] N. Seibe g and E. Wi en, “S ing heo y and noncommu a i e geome y,”
JHEP 09 (1999) 032, hep- h/9908142.
21
[4] H. G osse, C. Klimˇc´ık and P. P eˇsnajde , “Towa ds ini e quan um ield
heo y in noncommu a i e geome y,” In . J. Theo . Phys. 35 (1996) 231,
hep- h/9505175.
[5] U. Ca ow-Wa amu a and S. Wa amu a, “Chi ali y and Di ac ope a o on non-
commu a i e sphe e,” Commun. Ma h. Phys. 183 (1997) 365, hep- h/9605003;
“Noncommu a i e geome y and gauge heo y on uzzy sphe e,” Commun.
Ma h. Phys. 212 (2000) 395, hep- h/9801195.
[6] A. P. Balachand an, T. R. Go inda ajan and B. Yd i, “The e mion doubling
p oblem and noncommu a i e geome y,” Mod. Phys. Le . A 15 (2000) 1279,
hep- h/9911087; “The e mion doubling p oblem and noncommu a i e geom-
e y. II,” hep- h/0006216.
[7] H. G osse, C. Klimˇc´ık and P. P eˇsnajde , “Topologically non i ial ield con-
igu a ions in noncommu a i e geome y,” Commun. Ma h. Phys. 178 (1996)
507, hep- h/9510083; P. P eˇsnajde , “The o igin o chi al anomaly and he
noncommu a i e geome y,” J. Ma h. Phys. 41 (2000) 2789, hep- h/9912050.
[8] S. Baez, A. P. Balachand an, B. Id i and S. Vaidya, “Monopoles and soli ons
in uzzy physics,” Commun. Ma h. Phys. 208 (2000) 787, hep- h/9811169;
A. P. Balachand an and S. Vaidya, “Ins an ons and chi al anomaly in uzzy
physics,” In . J. Mod. Phys. A 16 (2001) 17, hep- h/9910129;
[9] M. Kon se ich, “De o ma ion quan isa ion o Poisson mani olds,”
q-alg/9709040.
[10] F. A. Be ezin, “Gene al concep o quan iza ion,” Commun. Ma h. Phys. 40
(1975) 153; “Quan iza ion in complex symme ic spaces,” Ma h. USSR, Iz . 9
(1975) 341.
[11] A. M. Pe elomo , Gene alized Cohe en S a es and hei Applica ions, Sp inge
(1986); “Cohe en s a es o a bi a y Lie g oups,” Commun. Ma h. Phys. 26
(1972) 222.
[12] B. Kos an , in G oup Rep esen a ions in Ma hema ics and Physics, ed.
C. T. Taam, Lec u e No es Phys, Vol. 6, Sp inge (1970), p. 237.
[13] A. A. Ki illo , Elemen s o he Theo y o Rep esen a ions, Sp inge (1976).
[14] J. Rawnsley, M. Cahen and S. Gu , “Quan iza ion o K¨ahle mani olds. I:
Geome ic in e p e a ion o Be ezin’s quan iza ion,” J. Geom. Phys. 7(1990)
45.
[15] M. Cahen, S. Gu and J. Rawnsley, “Quan iza ion o K¨ahle mani olds. II,”
T ans. Am. Ma h. Soc. 337 (1993) 73.
[16] A. V. Ka abego , “Pseudo-K¨ahle quan iza ion on lag mani olds,” Commun.
Ma h. Phys. 200 (1999) 355.
22
[17] H. Figue oa, J. M. G acia-Bond´ıa and J. V´a illy, “Moyal quan iza ion wi h com-
pac symme y g oups and noncommu a i e ha monic analysis,” J. Ma h. Phys.
31 (1990) 2664.
[18] A. P. Balachand an, B. Dolan, J. Lee, X. Ma in and D. O’Conno , “Fuzzy
complex p ojec i e spaces and hei s a -p oduc s,” hep- h/0107099.
[19] J. Mado e, “The uzzy sphe e,” Class. Quan . G a . 9(1992) 69.
[20] P. P eˇsnajde , “The o igin o chi al anomaly and he noncommu a i e geome-
y,” J. Ma h. Phys. 41 (2000) 2789, hep- h/9912050.
[21] Pe e P eˇsnajde , p i a e communica ion.
[22] J. Schi me , “A s a p oduc o complex G assmann mani olds,”
q-alg/9709021
[23] M. Bo demann, M. B ischle, C. Emm ich and S. Waldmann, “Subalgeb as wi h
con e ging s a p oduc s in de o ma ion quan iza ion: an algeb aic cons uc ion
o CPn,” J. Ma h. Phys. 37 (1996) 6311, q-alg/9512019; “Phase space e-
duc ion o s a -p oduc s: an explici cons uc ion o CPn,” Le . Ma h. Phys.
36 (1996) 357, q-alg/9503004.
[24] D. E. Li lewood, The Theo y o G oup Cha ac e s, Ox o d Uni e si y P ess
(1950).
[25] G. de B. Robinson, Rep esen a ion Theo y o he Symme ic G oup, Edinbu gh
Uni e si y P ess (1961).
[26] C. R. Hagen, A. J. Mac a lane, “Reduc ion o ep esen a ions o SUm+nwi h
espec o he subg oup SUm⊗SUn”J. Ma h. Phys. 6(1965) 1366.
[27] M. L. Whippman, ”B anching ules o simple Lie g oups” J. Ma h. Phys. 6
(1965) 1534.
[28] S. S e nbe g, G oup Theo y and Physics, Camb idge (1994).
[29] A. O. Ba u and R. R¸aczka, Theo y o G oup Rep esen a ions and Applica ions,
Wo ld Scien i ic (1986).
[30] B. P. Dolan, D. O’Conno and P. P esnajde , “Ma ix ϕ4models on he uzzy
sphe e and hei con inuum limi s,” hep- h/0109084.
[31] A. Y. Aleksee , A. Recknagel and V. Schome us, “Non-commu a i e wo ld-
olume geome ies: B anes on SU(2) and uzzy sphe es,” JHEP 09 (1999) 023,
hep- h/9908040.
[32] G. Felde , J. F ¨ohlich, J. Fuchs and C. Schweige , “The geome y o WZW
b anes,” J. Geom. Phys. 34 (2000) 162, hep- h/9909030.
23